\begin{answer}
    Note 
    $$
    \begin{aligned}
\theta_{\text{MAP}} &= \arg \max_\theta p(y|x, \theta)p(\theta)\\
&= \arg \max _\theta \log p(y|x, \theta) + \log (\theta)\\
&= \arg \max_\theta-\frac{1}{2\sigma^2} (X\theta - \vec{y})^T(X\theta - \vec{y}) - \frac{1}{2\eta^2}\theta^T\theta
\end{aligned}
$$

We take the derivative with respect to the objective and we get

$$
    \frac{1}{\sigma^2}X^T(\vec{y} - X\theta) - \frac{1}{\eta^2}\theta = 0
    $$

    or 

    $$
    \theta = (X^TX + \frac{\sigma^2}{\eta^2}I)^{-1}X^T\vec{y}
    $$
\end{answer}
